The butterfly stroke is hard on the muscles and also makes breathing difficult. The real problem lies in the coordination of movements in order to take a breath at the right time.
Air is taken in through the mouth, at the end of the pull-push phase, by lifting the head so as to look at the surface of the water. It must be fast so that the head immediately returns to its original place in order to keep the stroke as balanced as possible. For better "hydrodynamics", a breath is usually taken every other stroke. Exhaling however is continuous through the nose and mouth once the head is in the water.
The back crawl is a stroke that requires a lot of effort from the legs to stay buoyant and horizontal. The lower limbs are the ones consuming the most oxygen and that is why proper breathing is very important.
When on your back, your face is in the open air and oxygen is much easier to access. The difficulty here is more a matter of respiratory rate rather than the availability of air. According to the frequency of movement, the breathing cycle will be faster or slower. You must always fully exhale so that the next breath in through the mouth is more effective and provides the body with the maximum amount of oxygen.
We therefore breathe in during the arm recovery phase, and breathe out at the end of the recovery phase of the other arm. It's all about the rhythm!
The breaststroke is also known as the "frog stroke"!
The advantage of the breaststroke is that you breathe in with every movement, allowing a regular intake of air and making you out of breath less quickly.
Inhalation is through the mouth during the insweep phase of the arms as they push against the water and the shoulders rise above the water as the legs prepare to push. Like for the butterfly stroke, the head is raised so as to look in front of you and inhalation must be very brief. Exhalation, meanwhile, must be long and continuous throughout the underwater phase. Go make some waves!
The crawl breathing technique is undoubtedly the most difficult to master. It requires the head to be rotated to the side which can easily have a negative effect on balance and stroke technique.
Freestyle swimmers breathe in on the right or left side or by alternating the two. Again, it must be brief in order get the head pointing straight in the water as soon as possible.
However, there is no need to lift your whole face out of the water, you just need to get your mouth to the surface to be able to breathe "into the trough". Exhalation, always continuous, takes place when the head is back in the water.
Some prefer to swim the crawl by breathing every two strokes (unilateral) to fend off fatigue, others every three strokes (bilateral) to stay better aligned while swimming, or even more, for swimmers wanting to reduce their resistance in the water and therefore swim faster.